Boro goalkeeper Tomas Mejias saved from Rooney and Young, while Carrick blazed his effort over in between as Boro advanced to the quarter-finals with a 3-1 victory following a 0-0 draw after 120 minutes.
Louis van Gaal’s side were lacklustre on the night and fortunate to survive past 90 minutes, with Daley Blind twice surviving own-goal scares.
He saw one wild hack into the net chalked off for an offside flag against Kike before he breathed a huge sigh of relief as Sergio Romero’s mis-control from his back pass went narrowly wide.
Jesse Lingard struck a post in the 90th minute while Grant Leadbitter and Stewart Downing were denied by Romero in stoppage time.
The hosts should then have sealed their progress as they produced a spate of extra-time misses, Marouane Fellaini, twice, and substitute Anthony Martial guilty of missing headers.
And Boro made them pay on penalties as United joined Arsenal and Chelsea in crashing out at the last-16 stage.
The travelling contingent had early cause to cheer when George Friend nutmegged Lingard and, moments later, the same player should have sent them in ecstasy. United failed to clear a corner from the left and Emilio Nsue’s ambitious overhead kick fell straight into Friend’s path, but the full-back placed his effort directly at Romero.
Van Gaal had clearly seen enough at the interval, taking off James Wilson for Rooney, and the hosts at least tested Mejias with a series of shots down his throat, one from Memphis Depay that he almost fumbled back over his line.
That error would have made a blooper DVD but the one from Blind 12 minutes into the second period deserved to grace a cover.
Kike was flagged offside when hitting an effort against Romero’s far post and as the ball came back across the six-yard box, Blind somehow managed to slam the ball directly into the net with no one around him. His head entered his hands but fortunately for him the linesman’s flag spared his blushes.
The comedy of errors continued, though, as Romero allowed Blind’s back pass to almost creep in at the back post in a manner reminiscent of ex-Aston Villa goalkeeper Peter Enckelman’s howler against Birmingham.
That, coupled with Leadbitter’s fierce shot at Romero, prompted Van Gaal to call for Martial too in place of the disappointing Depay with 19 minutes left.
The game was meandering towards extra-time until both sides came to life in the dying stages. Lingard smashed an effort against a post before the visitors broke on the counter and Romero stood up to repel Leadbitter’s lob and then Downing’s drilled effort in stoppage time.
Martial could have prevented the flurry of spot-kicks that soon followed had he nodded in Lingard’s cross in.
Rooney was the first to miss and although David Nugent blazed over, Carrick did likewise before Mejias guessed right again to deny Young and send Boro through.
Romero, Darmian, Smalling, Blind, Rojo (Young 61), Carrick, Fellaini, Lingard, Andreas Pereira, Depay (Martial 71), Wilson (Rooney 46).
Mejias, Kalas, Stephens (Clayton 114), Ayala, Friend, Gibson, Leadbitter, Nsue, De Pena (Zuculini 79), Downing, Garcia (Nugent 84).
Lee Mason (Lancashire).
Middlesbrough v Everton, Southampton v Liverpool, Stoke v Sheffield Wednesday, Manchester City v Hull.